- #1

H_man

- 145

- 0

## Homework Statement

I wanted to show that a particular wave is composed of the 4 frequencies

W = W

_{0}+/- W

_{1}+/- W

_{2}

The equation for this FM wave is y(t) = Acos(W

_{0}t+W

_{1}COS(W

_{2}t)t )

I tried showing the different frequency components by performing a Fourier transform on the equation y(t). However I got a bit stuck with integrating the cosine term as part of the exponential function. I thought about expanding it as a Taylor series but that doesn't really help.

If this was a practical problem I would just use Matlab and a DFT and hey presto. But trying to do this analytically is posing a problem.

Expanding in terms of trigonometric identities to show the point also doesn't seem to get me closer to a solution.

y(t) = Acos(W

_{0}t+W

_{1}COS(W

_{2}t)t ) =

= A{cos(W

_{0}t)cos(W

_{1}COS(W

_{2}t)t ) -

sin(W

_{0}t)sin(W

_{1}sin(W

_{2}t)t )}

we are given that W

_{1}t<<1

and W

_{1}

^{2}t -> 0

so if we assume cos(W

_{1}COS(W

_{2}t)t )~1

Then we have:

= A{cos(W

_{0}t)-sin(W

_{0}t)W

_{1}cos(W

_{2}t)t )}

= A{cos(W

_{0}t) - sin(W

_{0}t+W

_{2}t)W

_{1}+

W

_{1}cos(W

_{0}t)sin(W

_{2}t)

But I don't see a solution from here...